Any number to the power of zero is 1. Meanwhile, zero to the power of any number is 0. Combination of these two rules makes the problematic of zero to the power of zero. If a0 = 1 and 0n = 0, then what would 00 be? In this post, we are going to explain the result of this expression.
Vlog
You can continue to read this tutorial or watch the following video. They both cover a gentle proof of why zeroth power of zero is equal to one.
An official guide to why zero-th power of zero is equal to one:
An unofficial guide to why zero-th power of zero is equal to one:
Let’s ask it to the python
The easiest way to reach to the answer is to use a calculator. On the other hand, the easiest way to reach to the answer for a software engineer is to use a programming language. Let’s ask zero to the zeroth power to python.
import math print(math.pow(0, 0))
The result is obviously seen to 1! This is acceptable because any number to the power of zero is 1. On the other hand, any number to the power of zero was 0. Now, it is confusing. Let’s try to find the reason of this answer and ask the question why.
An unofficial guide to proof of zero to the power of zero
The easiest way to explain of this expression is to use the definition of the limit. We know that one to the power of one is one as well.
11 = 1
Let’s re-define this expression as x to the power of x whereas x is equal to 1.
xx
Here, we are going to reduce the value of x with baby steps from 1 to something close to 0. Thereafter, we are going to monitor the result of this expression. This will show us the value result is closing to while input x is decreasing.
import math
xs = []; ys = []; index = []
x = 1
for i in range(0, 100):
x = x / 1.1
y = pow(x, x)
xs.append(x)
ys.append(y)
index.append(i)
So, we have the pair of input x and output y as series in the xs and ys list. Let’s plot the y values in a timeline. This will show us y goes to where while x is reducing.
import matplotlib.pyplot as plt plt.figure(figsize=(15, 7)) plt.xticks(index[::5], xs[::5], rotation='vertical') plt.plot(ys)
This graph basically shows the change of the value of “x to the power of x while x is going to zero“. x-axis represents x variables whereas y-axis represents the y value which is x to the power of x.
Its value seems decreasing and wants to go zero at the beginning but then it changes its idea and goes to 1 exactly 🙂 This explains why zero to zeroth power is actually one. But as I mentioned before, this is an unofficial way to explain the result. We still do not know why. Let’s deep dive into the proof!
An official guide to proof
We are going to use the official definition of limit to solve this problem. But before that we need to prove a couple of theorems.
Theorem 0 – log m (mn) = n
Let this be
log m (mn) = k
Exponential form of this expression is
mk = mn
If bases are equal, then exponents must be equal, too.
k = n
We supposed log m (mn) be k and we just found that k is equal to n.
log m (mn) = n
Theorem 1 – power property of logarithm: log b (xk) = k . log b x
b (xk) = k . log b x
Proof: y = log b x -> by = x
Let’s find the k-th power of terms in the both side.
(by)k = xk -> byk = xk
We can add the log base of b term in the both side.
log b (xk) = log b (byk)
b appears in both base and argument. So, this must be the power as proven in Theorem 0.
log b (xk) = yk
We supposed y was log b x in the first step of the proof of theorem 1.
log b (xk) = k.(log b x)
So, we proven the power property of logarithm!
Theorem 2: x = eln(x)
Proof: y = eln(x)
We can take the ln of two sides of the equation
ln(y) = ln(e)ln(x)
Let’s apply the power property of logarithm we just proven:
ln(y) = ln(x).ln(e) where ln(e) = 1
ln(y) = ln(x)
We can get rid of the ln term in the both of side. Therefore, y is equal to x.
y = x
y was eln(x) in the theorem 2. Replace it.
eln(x) = x
So, this is proven, too!
Theorem 3 – derivative of natural log of x is equal to 1 over x: (lnx)’ = 1/x.
Proof: y = lnx = log e x -> ey = x
Find the derivatives of both sides: (ey)’ = (x)’ = (ey)dy/dx = 1 -> dy/dx = 1/ey
Here, y was lnx: dy/dx = 1/ey = 1/elnx
Notice that we have proven e to the power of lnx is x in the theorem 2.
dy/dx = 1/elnx = 1/x. So, we can prove the derivative of lnx is 1/x.
Theorem 4: lim x->0 xx = 1
Here, I would like to replace x in the base as e to the power of ln(x) which is proven in theorem 2.
lim x->0 xx = lim x->0 (eln(x))x
Here, the base of this equation is e, and it is a constant number. We can apply the limit just in the power.
lim x->0 (eln(x))x = e ^ (lim x->0 x.lnx)
It’s better to remember the graph of lnx here.
Just focus on the power term. If we find the result of this power, we will be able to apply it to the base later.
lim x->0 x.lnx = 0 -∞
We can represent this as:
lim x->0 lnx / (1 / x) = -∞ / ∞
So, we have an intermediate form -∞/∞ and we can apply L’hospital rule here. In other words, we will be able to differentiate the numerator and denominator differently.
lim x->0 (lnx)’ / (1 / x)’
We have proven the derivative of lnx is equal to 1/x in the theorem 3.
lim x->0 (lnx)’ / (1 / x)’ = lim x->0 (lnx)’ / (x-1)’ = lim x->0 (1/x)(-1 x-2) = lim x->0 (1/x)(-1 / x2) = lim x->0 -x
To sum up, lim x->0 x.lnx = lim x->0 -x = 0
We have focused on just the power of term.
e ^ (lim x->0 x.lnx) = e ^ (0) = 1
So, this explains why 00 = 1!
A Second Proof – The Taylor Series for ex
Of course, the limit approach is not the only way to explain why 0 to the power of 0 is 1. We can get help from the Taylor series for ex.
ex = Σ (n=0 to ∞) xn / n! = x0 / 0! + x1 / 1! + x2 / 2! + x3 / 3! + …
Let’s find out the exact value of e0. We know that e is a constant value and any number to the power of zero is 1.
e0 = Σ (n=0 to ∞) 0n / n! = 00 / 0! + 01 / 1! + 02 / 2! + 03 / 3! + …
We know that zero to the power of any number is 0. Then, these values 01, 02, 03, … are going to be zero.
e0 = 00 / 0!
We already know those rules: zero factorial is equal to 1; and any non-zero number to the power of zero is 1. Replace these in the equation.
1 = 00 / 0!
1 = 00
So, we find the zero to the power of zero and it is equal to 1.
Conclusion
So, we have proven the result of zero to the zeroth power with many different ways. One is using calculator and math module of python programming language, second is using python to monitor the changes of x to the power of x values while x is closing to zero, third is using limit and derivative which requires a lot of math and calculus, and finally Taylor series for ex. I hope this tutorial helps you to understand why zero to the power of zero is one.
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